An Unconditionally Energy Stable Gradient Flow for Phase Field Modelling of Structural Topology Optimization in Geometrically Nonlinear Elasticity

We consider the topology optimization of geometrically nonlinear elasticity problems using a phase field model. A novel generalized stabilized semi-implicit scheme for the gradient flow is proposed to solve the resulting optimal control problem, thereby overcoming the need for extra adjoint variables and nonlinear constraints. Unconditional energy stability is shown for the Allen-Cahn type of gradient flow scheme in both continuous and discrete spaces. The local averaging and superconvergence patch recovery are applied to enhance the accuracy of the discrete gradient, thus the robustness of the phase field gradient flow for topology optimization in nonlinear elasticity. Numerical experiments show the effectiveness and robustness of the optimization algorithm proposed.